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probability and stochastic modeling

Fundamentals Of Probability With Stochastic Processes 4th Edition Saeed Ghahramani - Solutions

23. Let the joint probability density function of X and Y be given byShow that X and Y are dependent but E(XY ) = E(X)E(Y ). if xy, 0 < y < 1 f(x, y) = 0 otherwise.
22. Let X and Y be two independent random integers from the set {0, 1, 2, . . . , n}. Find P(X = Y ) and P(X ≤ Y ).
21. The lifetimes of mufflers manufactured by company A are random with the following probability density function:The lifetimes of mufflers manufactured by company B are random with the following probability density function:Elizabeth buys two mufflers, one from company A and the other one from
20. Six brothers and sisters who are all either under 10 or in their early teens are having dinner with their parents and four grandparents. Their mother unintentionally feeds the entire family (including herself) a type of poisonous mushrooms that makes 20% of the adults and 30% of the children
19. A point is selected at random from the diskLet X be the x-coordinate and Y be the y-coordinate of the point selected. Determine if X and Y are independent random variables. R={(x,y) = R2: x + y
18. LetX and Y be independent randompoints from the interval (0, 1). Find the probability density function of the random variable XY .
17. Let X and Y be independent random points from the interval (−1, 1). Find[Emax(X, Y )].
16. Let X and Y be independent exponential random variables both with mean 1. Find[Emax(X, Y )].
15. Let X and Y be two independent random variables with the same probability density function given byShow that g, the probability density function of X/Y, is given by f(x) = 0 if 0 < x < elsewhere.
14. Let the joint probability density function of X and Y be given byFind E(X2Y ). 2e-(2+2y) f(x,y) = if x 0, y0 0 otherwise.
13. Let the joint probability density function of X and Y be given byDetermine if E(XY ) = E(X)E(Y ). 8xy if 0
12. Let the joint probability density function of random variables X and Y be given byAre X and Y independent?Why or why not? f(x, y) = Le-(y+1) if x 0, y 0 elsewhere.
11. Suppose that the amount of cholesterol in a certain type of sandwich is 100X milligrams, where X is a random variable with the following probability density function:Find the probability that two such sandwiches made independently have the same amount of cholesterol. f(x)= 2x+3 if 2
10. Let the joint probability density function of random variables X and Y be given byAre X and Y independent?Why or why not? 2 if 0 y x1 f(x, y) = 0 elsewhere.
9. The joint probability mass function p(x, y) of the random variables X and Y is given by the following table. Determine if X and Y are independent. Y H 0 1 2 3 0 0.1681 0.1804 0.0574 0.0041 1 0.1804 0.1936 0.0616 0.0044 2 0.0574 0.0616 0.0196 0.0014 3 0.0041 0.0044 0.0014 0.0001
8. A fair coin is tossed n times by Adam and n times by Andrew. What is the probability that they get the same number of heads?
7. Let X and Y be two independent random variables with distribution functions F and G, respectively. Find the distribution functions of max(X, Y ) and min(X, Y ).
6. Suppose that the number of claims received by an insurance company in a given week is independent of the number of claims received in any other week. An actuary has calculated that the probability mass function of the number of claims received in a random week isFind the probability that the
3. LetX and Y be independent randomvariables each having the probabilitymass functionFind P(X = 1, Y = 3) and P(X + Y = 3). p(x) 23 2\3 H x = 1,2,3,...
2. Let the joint probability mass function of random variables X and Y be given byAre X and Y independent?Why or why not? xy if (x, y) (1, 1), (1, 2), (2, 1) p(x,y) = ==== elsewhere.
1. Let the joint probability mass function of random variables X and Y be given byAre X and Y independent?Why or why not? -(x + y) if x 1,2, y = 0,1,2 25 p(x,y) elsewhere.
Prove that two random variables X and Y with the following joint probability density function are not independent. 8xy f(x,y) 0 x y 1 otherwise.
A plane is ruled with parallel lines a distance d apart. A needle of length ℓ, ℓ
Stores A and B, which belong to the same owner, are located in two different towns. If the probability density function of the weekly profit of each store, in thousands of dollars, is given byand the profit of one store is independent of the other, what is the probability that next week one store
A point is selected at random from the rectangleLet X be the x-coordinate and Y be the y-coordinate of the point selected. Determine if X and Y are independent random variables. R={(x,y) R2:0
Suppose that 4% of the bicycle fenders, produced by a stamping machine from the strips of steel, need smoothing. What is the probability that, of the next 13 bicycle fenders stamped by this machine, two need smoothing and, of the next 20, three need smoothing?
2. Let X be a car insurance company’s annual losses under medical coverage insurance.Let Y be its annual losses under comprehensive and collisions losses. Suppose that the joint probability density function of X and Y, in millions of dollars, is(a) Find the marginal probability density function
1. A device with two components functions only if both components function. Suppose that the joint probability density function of X and Y, the lifetimes of the components, in years, is(a) Find the probability that the device functions for at least 2 years.(b) Find the marginal distribution
36. As Liu Wen from Hebei University of Technology in Tianjin, China, has noted in the April 2001 issue of The American Mathematical Monthly, in some reputable probability and statistics texts it has been asserted that “if a two-dimensional distribution function F(x, y) has a continuous density
35. For α > 0, β > 0, and γ > 0, the following function is called the bivariate Dirichlet probability density functionif x ≥ 0, y ≥ 0, and x + y ≤ 1; f(x, y) = 0, otherwise. Prove that fX, the marginal probability density function of X, is beta with parameters (α, β + γ); and
34. Consider a disk centered at O with radius R. Suppose that n ≥ 3 points P1, P2, . . . , Pn are independently placed at random inside the disk. Find the probability that all these points are contained in a closed semicircular disk.Hint: For each 1 ≤ i ≤ n, let Ai be the endpoint of the
33. Let X and Y be continuous random variables with joint probability density function f(x, y). Let Z = Y/X, X 6= 0. Prove that the probability density function of Z is given by fz(2) = = |x|f(x, xz) dx.
32. A point is selected at random and uniformly from the region R =(x, y) : |x| + |y| ≤ 1.Find the probability density function of the x-coordinate of the point selected at random.
31. Two points are placed on a segment of length ℓ independently and at random to divide the line into three parts.What is the probability that the length of none of the three parts exceeds a given value α, ℓ/3 ≤ α ≤ ℓ?
30. A farmer who has two pieces of lumber of lengths a and b (a
29. Two numbers x and y are selected at random from the interval (0, 1). For i = 0, 1, 2, determine the probability that the integer nearest to x + y is i.Note: This problem was given by Hilton and Pedersen in the paper “A Role for Untraditional Geometry in the Curriculum,” published in the
28. Three points M, N, and L are placed on a circle at random and independently.What is the probability thatMNL is an acute angle?
27. Let g and h be two probability density functions with distribution functions G and H, respectively. Show that for −1 ≤ α ≤ 1, the function f(x, y) = g(x)h(y)????1 + α[2G(x) − 1][2H(y) − 1]is a joint probability density function of two random variables. Moreover, prove that g and h
26. Let X and Y be random variables with finite expected values. Show that if P(X ≤ Y ) = 1, then E(X) ≤ E(Y ).
25. Suppose that h is the probability density function of a continuous random variable. Let the joint probability density function of two random variables X and Y be given by f(x, y) = h(x)h(y), x ∈ R, y ∈ R.Prove that P(X ≥ Y ) = 1/2.
24. Let X and Y have the joint probability density functionFind the joint distribution function, the marginal distribution functions, and the marginal probability density functions of X and Y . f(x, y): = 0 (x + y) 0
23. LetDetermine if F is the joint distribution function of two random variables X and Y . 20 or y 0 xy(x + y) 0 <
22. For λ > 0, letDetermine if F is the joint distribution function of two random variables X and Y . F(x, y) = 1- Ae-(x+y) if x > 0, y > 0 otherwise.
21. Two points X and Y are selected at random and independently from the interval (0, 1).Calculate P(Y ≤ X and X2 + Y 2 ≤ 1).
20. On a line segment AB of length ℓ, two points C and D are placed at random and independently.What is the probability that C is closer to D than to A?
19. A farmer makes cuts at two points selected at random on a piece of lumber of length ℓ.What is the expected value of the length of the middle piece?
18. A man invites his fianc´ee to an elegant hotel for a Sunday brunch. They decide to meet in the lobby of the hotel between 11:30 A.M. and 12 noon. If they arrive at random times during this period, what is the probability that the first to arrive has to wait at least 12 minutes?
17. Let R be the bounded region between y = x and y = x2. A random point (X, Y ) is selected from R.(a) Find the joint probability density function of X and Y .(b) Calculate the marginal probability density functions of X and Y .(c) Find E(X) and E(Y ).
16. Let F be the joint distribution function of the random variablesX and Y . In terms of F, calculate P(X > t, Y > u).
15. Let F be the joint distribution function of random variables X and Y . For x1 < x2 and y1 < y2, in terms of F, calculate P(x1 < X ≤ x2, y1 < Y ≤ y2).
14. Let X be the proportion of customers of an insurance company who bundle their auto and home insurance policies. Let Y be the proportion of customers who insure at least their car with the insurance company. An actuary has discovered that for, 0 ≤ x ≤ y ≤ 1, the joint distribution function
13. Let X and Y have the joint probability density functionCalculate P(X + Y ≤ 1/2), P(X − Y ≤ 1/2), P(XY ≤ 1/4), and P(X2 + Y 2 ≤ 1). if 0x1, 0 y 1 f(x, y) = elsewhere.
12. Let the joint probability density function of random variables X and Y be given byFind the marginal probability density functions of X and Y . ye if x>0, 0
11. Let the joint probability density function of random variables X and Y be given by(a) Calculate the marginal probability density functions of X and Y, respectively.(b) Calculate E(X) and E(Y ). 8xy if 0 y x1 f(x, y) = 0 elsewhere.
10. Let the joint probability density function of random variables X and Y be given by(a) Calculate the marginal probability density functions of X and Y, respectively.(b) Find E(X) and E(Y ).(c) Calculate P(X 2 f(x, y) if 0 y x1 elsewhere.
9. From an ordinary deck of 52 cards, seven cards are drawn at random and without replacement.Let X and Y be the number of hearts and the number of spades drawn, respectively.(a) Find the joint probability mass function of X and Y .(b) Calculate P(X ≥ Y ).
8. In an area prone to flood, an insurance company covers losses only up to three separate floods per year. Let X be the total number of floods in a random year and Y be the number of the floods that each cause over $74 million in insured damage. Suppose that, for some constantc, the joint
7. In a community 30% of the adults are Republicans, 50% are Democrats, and the rest are independent. For a randomly selected person, letCalculate the joint probability mass function of X and Y . X = = 1 Y 0 if he or she is a Republican otherwise, if he or she is a Democrat otherwise.
4. Let the joint probability mass function of discrete random variables X and Y be given byFind P(X > Y ), P(X + Y ≤ 2), and P(X + Y = 2). (x + y) if x 1,2, y = 0,1,2 25 p(x, y) otherwise.
3. Let the joint probability mass function of discrete random variables X and Y be given byDetermine (a) the value of the constant k, (b) the marginal probability mass functions of X and Y, and (c) E(X) and E(Y ). p(x, y) = = (k(x + y) 0 if (x, y) = (1, 1), (1, 3), (2, 3) otherwise.
2. Let the joint probability mass function of discrete random variables X and Y be given byDetermine (a) the value of the constantc, (b) the marginal probability mass functions of X and Y, (c) P(X ≥ 2 | Y = 1), (d) E(X) and E(Y ). p(x, y) [c(x + y) if x 1,2,3, y= 1,2 = otherwise.
1. Let the joint probability mass function of discrete random variables X and Y be given byDetermine (a) the value of the constant k, (b) the marginal probability mass functions of X and Y, (c) P(X > 1 | Y = 1), (d) E(X) and E(Y ). p(x, y) = if x 1,2, y = 1,2 otherwise.
Let X and Y have joint probability density functionFind E(X2 + Y2). f(x, y) = 32 (x + y). if 0 <
A farmer decides to build a pen in the shape of a triangle for his chickens.He sends his son out to cut the lumber and the boy, without taking any thought as to the ultimate purpose, makes two cuts at two points selected at random. What are the chances that the resulting three pieces of lumber can
A man invites his fianc´ee to a fine hotel for a Sunday brunch. They decide to meet in the lobby of the hotel between 11:30 A.M. and 12 noon. If they arrive at random times during this period, what is the probability that they will meet within 10 minutes?
A circle of radius 1 is inscribed in a square with sides of length 2. A point is selected at random from the square.What is the probability that it is inside the circle? Note that by a point being selected at random from the square we mean that the point is selected in a way that all the subsets of
For λ > 0, letDetermine if F is the joint distribution function of two random variables X and Y . F(x, y) -{ 1- Ae-(x+y) if x>0, y> 0 otherwise.
The joint probability density function of random variables X and Y is given by(a) Determine the value of λ.(b) Find the marginal probability density functions of X and Y .(c) Calculate E(X) and E(Y ) (Axy f(x, y) 0 0x y 1 otherwise.
Let the joint probability mass function of the discrete random variablesX and Y be given byFind E(XY ). () if x=1,2, y = 1,2,3 p(x, y) = otherwise.
Let the joint probability mass function of random variables X and Y be given byFind E(X) and E(Y ). p(x, y) 70 x(x+y) if x = 1,2,3, y=3,4 0 elsewhere.
Roll a balanced die and let the outcome be X. Then toss a fair coin X times and let Y denote the number of tails. What is the joint probability mass function of X and Y and the marginal probability mass functions of X and Y ?
A small college has 90 male and 30 female professors. An ad hoc committee of five is selected at random to write the vision and mission of the college. Let X and Y be the number of men and women on this committee, respectively.(a) Find the joint probability mass function of X and Y .(b) Find pX and
10. A wireless phone company manufactures cell phone chips for 40% of the cell phones it produces and purchases chips for the remaining 60% of its cell phone production from a supplier. Suppose that the lifetime of a chip manufactured by the company’s own plant is gamma with mean 4 and variance 4
9. In a mayoral election in a small island in Greece, of 5000 voters only 1000 voted for Tasoula. What is the probability that, in a random sample of 100 voters, more than 15 voted for this mayoral candidate?
8. To improve the reliability of a system, sometimes manufacturers add one or more operating components in parallel so that if one of the components fails, the system still continues to function. Therefore, a parallel system functions if and only if at least one of its components functions. Suppose
7. LetX be the weight of a randomly selected adult from a certain community, and assume that X is a normal random variable. If it is known that, in that community, the mean of the weights of the adults is 160 pounds, and 85% of the adult population has a weight between 120 and 200 pounds, find
6. Suppose that, every day, all over the world, 10 million people search the Web for film review websites. Suppose that, among many other sites, the search always turns up a specific website that aggregates reviews of movies and TV shows, independently of all the other sites. If 0.2% of such people
5. Suppose that buses arrive at a station every 15 minutes. During the operating hours of the buses, Kayla arrives at the station at a random time. (a) If she has been waiting for the bus for 9 minutes, what is the probability that she has to wait at least another 4 minutes?(b) Calculate the same
4. In a huge office building, the alarm system has n sensors, the lifetime of each being exponential with mean 1/λ, independently of the other ones. If for the next t units of time the alarm system is not checked, what is the probability that at time t only k sensors are working? Assume that
2. While in flight, the fraction X of the time that Nolan sleeps is beta with parameters(6, 10). What is the probability that next time Nolan flies from Chicago to Shanghai, he sleeps longer than 6 hours during the 14-hour flight?
1. In a city, a passenger transit bus operates two routes that both stop at the Downtown Center and at the Embarcadero stations. Route B1 operates every day between 6:00 A.M.and 9:30 P.M. and arrives at Embarcadero every 15 minutes starting at 6:15 A.M. Route B26 operates during the same period but
15. Let X be a uniform random variable over the interval (1 − θ, 1 + θ), where 0 < θ < 1 is a given parameter. Find a function ofX, say g(X), so thatEg(X)= θ2.
10. Determine the value(s) of k for which the following is a probability density function.f(x) = ke−x2+3x+2, −∞ < x < ∞.
6. Let X be a uniform random variable over the interval (0, 1). Calculate E(−lnX).
5. The time that it takes for a computer system to fail is exponential with mean 1700 hours.If a lab has 20 such computer systems, what is the probability that at least two fail before 1700 hours of use?
4. Let X, the lifetime of a light bulb, be an exponential random variable with parameter λ.Is it possible that X satisfies the following relation?P(X ≤ 2) = 2P(2 < X ≤ 3).If so, for what value of λ?
2. The hazard function of a random variable X with the set of possible values (0, 1), is given byFind the probability density function of X. x(t): 2 1-t 0
1. A batch of cell phones manufactured by a wireless company’s plant in the U.S.A. has been ordered by the company’s main facility in Portugal. Suppose that the hazard function of the delivery lead time, in weeks, is λ(t) = 1/8, t ≥ 0. If the order for the batch of cell phones was placed
5. One of the most popular distributions used to model the lifetimes of electric components is theWeibull distribution, whose probability density function is given byDetermine for which values of α the hazard function of a Weibull random variable is increasing, for which values it is decreasing,
3. Suppose that the lifetime of a cell phone chip manufactured by a wireless company, in years, is gamma with parameters r = 4 and λ = 1. Find λ(t), the hazard function of such a chip, and its instantaneous failure rate if it has survived for four years. Using this, find an approximate value for
2. The hazard function of a random variable X with the set of possible values [1, 3), is given byFind the probability density function of X. 3t x(t) 1
1. Suppose that the lifetime of a device, in hundreds of hours, is a random variable with probability density function(a) Find λ(t), the hazard function of the lifetime of the device.(b) Find the instantaneous failure rates of such a device if it has survived 50 hours, 90 hours, and 99 hours,
2. LetX be a randomnumber between 0 and 1. Show that Y = X2 is beta with parameters 1/2 and 1.
1. Suppose that each time the price of a commodity decreases,X, the proportion of change in the price, is a beta random variable with parametrs α = 1 and β = 10. That is, if, the price of commodity changes, say, from q1 to q2, q2 < q1, then X = (q1 − q2)/q1 is beta with parameters 1 and 10.
13. For an integer n ≥ 3, let X be a random variable with the probability density functionSuch random variables have significant applications in statistics. They are called t-distributed with n degrees of freedom. Using the previous two exercises, find E(X)and Var(X). f(x) = r(n+1) / (1+2)
12. Prove that B(, B) (a)(3) T(a+B)
11. For α, β > 0, show thatHint: Make the substitution x = t2/(1 + t2) in B(a,b) = 2 = 2 for 2a-1 (1 + ()-(a+8) dt.
10. Under what conditions and about which point(s) is the probability density function of a beta random variable symmetric?
3. For what value of c is the following a probability density function of some random variable X? Find E(X) and Var(X). f(x)= Scr (1-x) 0 < x
2. Is the following a probability density function?Why or why not? 120x2(1-x) 0

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